Simultaneous Price Setting

• Price Competition: Unlike Cournot, where firms compete by choosing quantities, in the Bertrand model, firms compete by setting prices and letting the market determine the quantity sold. Simultaneous, non-cooperative decisions.
• Homogeneous Products: Firms produce identical products, making price the sole factor for consumers when choosing between firms
• Cost Structures: Firms have identical cost structures and sufficient capacity to meet all demand
• Rational Expectations: Firms are profit-maximizers and have rational expectations about their competitors’ behavior

Bertrand Model

• Profit Maximization for Firm 1: \[ \pi_1 = P_1 \cdot Q_1(P_1, P_2) -C(Q_1(P_1,P_2)) \]
  • where \(P_1\) is the price set by Firm 1,
  • \(C(Q_1(P_1,P_2))\) is cost function for Firm 1, and
  • \(Q_1(P_1, P_2)\) is the demand for Firm 1’s product as a function of both firms’ prices

Demand Scenarios for Firm 1

1. When \(P_1 = P_2\):
   • If both firms set equal prices, we assume demand is split equally between them. Hence, \(Q_1 =Q_2 = \frac{1}{2}Q(P_1)\), where \(Q(P_1)\) is the total market demand at price \(P_1\)
2. When \(P_1 > P_2\):
   • Firm 1 captures none of the market demand since consumers prefer the cheaper, identical product from Firm 2. Thus, \(Q_1 = 0\)
3. When \(P_1 < P_2\):
   • Firm 1 captures the entire market demand because its price is lower. Hence, \(Q_1 = Q(P_1)\)

Residual Demand Bertrand

We can summarize the previous analysis as \[ Q_1(P_1,P_2) = \begin{cases} 0 ,& P_1>P_2 \\ \frac{1}{2}Q(P_1),& P_1=P_2 \\ Q(P_1) ,& P_1 < P_2 \end{cases} \]

Residual Demand Bertrand

Example (continued)

Bertrand Equilibrium

• Definition: The Bertrand equilibrium is reached when each firm sets its price such that it maximizes its profits and neither firm can increase profits by unilaterally changing its price.

• Reaching the Bertrand Equilibrium:

  • With identical products and rational, profit-maximizing firms, the equilibrium occurs when both firms set their prices equal to marginal cost (\(P = MC\)).
  • If one firm undercuts the other by even a small amount, it captures the entire market. Thus, to avoid losing the market, firms race to the bottom, stopping at \(P = MC\).

Bertrand Equilibrium

• Characteristics:
  • The Bertrand equilibrium results in the competitive price level, even with only two firms in the market.
  • Total market supply meets all demand at this price level, as firms have sufficient capacity.

Bertrand vs. Cournot

• In Cournot competition, firms choose quantities,
  • leading to higher prices than marginal costs, \(P>MC\), due to the strategic reduction in total output.
• In Bertrand competition, firms choose prices,
  • leading to a price equal to marginal cost, \(P=MC\), similar to perfect competition outcomes, even with only two firms.

Bertrand and Perfect Competition

• Both result in prices equal to marginal costs, \(P=MC\);
• however, perfect competition assumes an infinite number of firms, \(N=\infty\),
• while Bertrand demonstrates that two firms \(N\ge2\), are sufficient to reach competitive pricing under price competition.

Quantity Leadership

• Sequential Moves: Unlike Cournot and Bertrand models, the Stackelberg model assumes that firms move sequentially. One firm, designated as the leader, chooses its output first; the follower firm then reacts to this decision.
• Homogeneous Products: Firms produce identical products.
• Known Reaction Function: The leader firm is aware of the follower firm’s reaction function and uses this information to optimize its own output decision.
• Complete Information: Both firms have complete information about the market, including costs and demand.

The Follower’s Problem

• General Framework: The follower firm chooses its output \(Q_2\) to maximize its profit, given the output \(Q_1\) of the leader firm. The follower’s profit function is: \[ \pi_2(Q_2) = P(Q_1 + Q_2)Q_2 - C_2(Q_2) \] where \(P(Q)\) is the market price function and \(C_2(Q_2)\) represents the cost of producing \(Q_2\).

Deriving the Reaction Function

• The profit-maximization choice of the follower firm will depend on the leader’s choice \(Q_1\)
• The reaction function \(Q_2 = BR_2(Q_1)\) shows this relationship
• \(BR_2(Q_1)\) derived by solving: \[ \frac{d\pi_2}{dQ_2} = 0 \] This equation gives the best response of the follower to any given \(Q_1\) by the leader.

Follower’s Reaction Curve

• Assumptions: Linear inverse demand curve \(P(Q) = a - b(Q_1 + Q_2)\) and constant marginal costs \(MC_1 = MC_2 = c\).

• Follower’s Reaction Function: With these assumptions, the follower’s reaction function simplifies to: \[ Q_2 = \frac{a - c - bQ_1}{2b} \qquad(2)\] This function indicates how the follower’s optimal output varies with the leader’s output level.

Isoprofit Curves

• Isoprofit curves represent combinations of \(Q_1\) and \(Q_2\) that yield the same profit level for the follower. These curves help illustrate the follower’s best response to different leader outputs.

• In the case of linear demand and constant marginal costs, isoprofit curves are comprised of all points (\(Q_1,Q_2\)) that satisfy equations of the form \[ \bar\pi_2=(a-c)Q_2-bQ_1Q_2-bQ_2^2 \]

Reaction and Isoprofit Curves

Reaction and Isoprofit Curves

• Profits of firm 2 will increase as we move to isoprofit lines that are closer to the left, (\(Q_1\) decreases)
• Highest profits when firm 2 is a monopolist (\(Q_1=0\))
• Profit-maximization implies that for each choice of \(Q_1\), firm 2 will pick the value of \(Q_2\) that reaches the isoprofit curve furthest to the left
• It follows that the slope of the isoprofit line must be vertical at the optimal choice (tangency condition)
• The locus of these tangencies form \(BR_2(Q_1)\)

The Leader’s Problem

• Leader’s Objective: Given the follower’s reaction function, the leader firm optimizes: \[ \begin{aligned} \max_{Q_1}\pi_1(Q_1) &= P(Q_1 + Q_2)Q_1 - C_1(Q_1) \\ &\text{s.t. } Q_2=BR_2(Q_1) \end{aligned} \]

• The leader internalizes that its actions influence the output of the follower

  • total output \(Q=Q_1+BR_2(Q_1)\)

The Leader’s Problem

• With linear demand and constant marginal costs \[ \begin{aligned} \max_{Q_1}\pi_1(Q_1) &= (a - b(Q_1 + Q_2))Q_1 - cQ_1 \\ &\text{s.t. } Q_2=\frac{a - c - bQ_1}{2b} \end{aligned} \]

The Leader’s Problem

• Substituting the follower’s reaction function into the leader’s profit function and simplifying we get \[ \max_{Q_1}\pi(Q_1)=\frac{a-c}{2}Q_1-\frac{b}{2}Q_1^2 \]

Optimal Quantities

• Solving the leader’s maximization problem yields the Stackelberg equilibrium quantities for both firms:
• For the leader \(Q_1^*=\frac{a-c}{2b}\)
• and the follower, substituting into the \(BR_2(Q_1),\) \(Q_2^*=\frac{a-c}{4b}\)
• Total output \(Q^*=\frac{3(a-c)}{4b}\)
• Market price \(P^*(Q)=\frac{a+3c}{4}\)

Stackelberg Equilibrium

Stackelberg Equilibrium

• The follower will choose an output along its reaction curve \(BR_2(Q_1)\)
• The leader chooses an output combination that yields the highest profits
• Thus, the leader chooses output such that the isoprofit curve is furthest down and tangent to the firm’s 2 best reaction curve
• In other words, firm 1 selects the point along \(BR_2(Q_1)\) that reaches the highest isoprofit curve;
• The reaction curve is tangent to the isoprofit curve at this point

Discussion

• The leader’s advantage of moving first allows it to secure a more favorable portion of the market, influencing the overall market structure and outcomes
• Unlike the Cournot outcome, which was symmetric, under the Stackelberg outcome, the leader produces more output than the follower (exactly twice as much in fact).
• Even though the market price is lower under the Stackelberg outcome than under the Cournot outcome, the leader’s profit under the Stackelberg outcome is higher than its profit at the Cournot equilibrium.
• An oligopolist benefits by choosing its output first (sequential game)

Price Leadership Model

• Dominant Firm:
  • One firm has a large enough market share to set the price. This firm is the price leader.
  • The dominant firm considers the residual demand curve, which is the market demand curve minus the supply of the competitive fringe.
• Follower Firm:
  • In equilibrium, the follower must always set the same price as the leader (follows from homogeneous product assumption)
  • Followers take the price set by the dominant firm and choose their profit-maximizing output (competitive behavior)

The Follower’s Problem

• Objective: The follower firm aims to maximize its profits given the price set by the leader. The follower assumes this price as fixed and decides its quantity accordingly.

• Profit Maximization: \[ PQ_2 - C_2(Q_2) \] where \(P\) is the price set by the leader, and \(C_2(Q_2)\) is the cost function for the follower.

• Output Decision: The follower determines its output level \(Q_2\) where price equals marginal cost (\(P = MC_2\)), leading to a supply curve for the follower \(S(P)\).

The Leader’s Problem

• Forecasting Follower’s Response: The leader sets a price \(P\) anticipating the follower’s response. The leader must solve for the price that maximizes its profit, considering the residual demand it faces after the follower’s supply.

• Residual Demand Curve: The leader’s demand is the market demand minus the follower’s supply, \(R(P) = D(P) - S(P)\).

• Profit Maximization for the Leader: The leader’s profit is \(\pi_1(P) = (P - c)[D(P) - S(P)] = (P - c)R(P)\). The leader aims to choose \(P\) where marginal revenue from the residual demand curve equals marginal cost.

Example

• Demand is \(D(P) = \frac{a}{b}- \frac{P}{b}\),
• the follower’s cost function is quadratic, \(C_2(Q_2)=Q^2_2/2\) and
• the leader’s cost function is \(C_1(Q_1)=cQ_1\)
• Solving the follower’s problem we can derive the follower’s supply curve \[ P=Q_2 \implies S(p)=P \]

Example

• Residual demand for the leader \[ R(P)=D(P)-S(P)=\frac{a}{b}- \frac{P}{b}-P=\frac{a}{b}- \frac{b+1}{b}P \]
• Solving for P, \(P=\frac{a}{b+1}-\frac{b}{b+1}Q_1\)
• Leader’s Optimal Price and Output: By setting marginal revenue equal to marginal cost for the residual demand curve, \(Q_1^*=\frac{a-c(b+1)}{2b}\); \(P^*=\frac{a-c(b+1)}{2(b+1)}\) using the residual demand curve (!!)

Equilibrium

Competitive Fringe Growth

Differentiated Goods

• Differentiated products: products that consumers consider distinctive from each other.

• Vertical differentiation is about inferiority or superiority.

  • Two products are vertically differentiated when consumers consider one product better or worse than the other.
  • Duracell batteries are vertically differentiated from generic store-brand batteries because they last longer.

Horizontal Differentiated Goods

• Horizontal differentiation is about substitutability.
  • Two products, A and B, are horizontally differentiated when, at equal prices, some consumers view B as a poor substitute for A and thus will buy A even if A’s price is higher than B’s, while other consumers view A as a poor substitute for B and thus will buy B even if B’s price is higher than A’s.
  • Diet Coke and Diet Pepsi are horizontally differentiated.
• Horizontal differentiation is an important concept for the theory of oligopoly and monopolistic competition

Horizontal Differentiation

• Firms selling horizontally differentiated products have downward-sloping demand curves.

• Weak horizontal differentiation: the firm’s demand is quite sensitive to a change in its own price and the prices of its rivals.

• Strong horizontal differentiation: the firm’s demand is much less sensitive to a change in its own price and the prices of its rivals.

Horizontal Differentiation

Price decrease of competitor -> Shifts \(D\) to \(D'\)

Bertrand with Differentiated Products

• Firms have horizontally differentiated products and compete in prices
• Demand for firm 1 \(Q_1(P_1,P_2)\) (and firm 2) are now continuous (we can take derivatives now!) \[ \begin{aligned} \frac{\partial Q_1(P_1,P_2)}{\partial P_1} &<0 \\ \frac{\partial Q_1(P_1,P_2)}{\partial P_2} &>0 \end{aligned} \]

Profit-Max Firms’ Problem

• For firm 1, \[ \max_{P_1} \pi(P_1)= P_1Q_1(P_1,P_2)-C_1(Q_1(P_1,P_2)) \]

• where \(P_i\) is the price of firm \(i=1,2\), \(Q_1(P_1,P_2)\) is the demand of firm 1, and \(C_1\) is the cost function of firm 1

Price Reaction Function

• As in the Cournot model, an equilibrium occurs when each firm is doing the best it can given the actions of its rival.
• We begin by deriving each firm’s price reaction function—its profit-maximizing price as a function of its rival’s price.
• Deriving with respect to \(P_1\) and solving for \(P_1\), \[ P_1=R_1(P_2) \]
• We can do the same for firm 2, \(P_2=R_2(P_1)\)

Price Reaction Curves

Example of reaction functions. The Bertrand equilibrium occurs where the two reaction functions intersect, E. This differs from the monopoly equilibrium, M.

Equilibrium

• Product differentiation softens price competition.
  • When products are as strongly differentiated, price cutting is less effective for stealing a rival’s business than when products are perfect substitutes.
  • However, firms incur a heavy cost to achieve this product differentiation.
• The Bertrand equilibrium prices do not correspond to the monopoly prices
  • As in the Cournot model, independent profit-maximizing oligopolists will typically not attain the monopolist outcome, because neither firm takes into account the adverse/beneficial effect that a price cut/increase would have on its rival.

Monopolistic Competition

• Three distinguishing features.

1. Fragmented market—it consists of many buyers and sellers.
2. Free entry (and exit)—any firm can hire the inputs (labor, capital, and so forth) needed to compete in the market, and they can release these inputs from employment when they do not need them.
3. Firms produce horizontally differentiated products—consumers view firms’ products as imperfect substitutes for each other.

• Market fragmentation and free entry and exit are also characteristics of perfectly competitive markets.

Equilibrium

• In choosing their prices, monopolistic competitors behave like the differentiated products oligopolists
• Even though the market is fragmented, each firm’s demand curve is downward sloping because of product differentiation.
• Taking the prices of other firms as given, each firm maximizes its profit at the point at which its marginal revenue equals marginal cost
• As in the Bertrand model of oligopoly with differentiated products, the market attains an equilibrium when every firm is charging a price that is the best response to the set of prices charged by all other firms in the market.

Short-Run Equilibrium

• The key difference between monopolistic competition and differentiated products oligopoly is free entry.
  • If there are profit opportunities in the market, new entrants will appear to seize them
• Hence, in the short run profits might be positive

Short-Run Equilibrium

Long-Run Equilibrium

• If profits are positive in the short-run, more firms will enter the market to take advantage of the profit opportunity.
• As more firms come into the market, each firm’s share of overall market demand will fall—the typical firm’s demand curve will shift leftward.
• Entry and the resultant leftward shift in firms’ demand curves will cease when firms make zero economic profit.

Long-Run Equilibrium

Possible Long-Run Equilibrium Prices and Quantities

• Even though in monopolistic competition long-run equilibrium profits are zero, the equilibrium price and quantities might be lower or higher
• Long-run equilibrium prices and quantities will depend on the price elasticity of the demand curve, and or the average cost curve

Long-Run Equilibrium - Price Elasticity of Demand

Long-Run Equilibrium - Price Elasticity of Demand

• In a long-run equilibrium with highly elastic demand,
  • the margin (P* − MC) between price and marginal cost is small, and
  • firms produce a large volume of output.
• In a long-run equilibrium with inelastic demand,
  • the margin between price and marginal cost is large and
  • each firm produces a small volume of output.
• If the total number of units purchased in equilibrium is the same in both, Marketa A and B, Market B would have more firms because each firm sells a smaller quantity than each firm in Market A.

Long-Run Equilibrium - Shift in AC

Long-Run Equilibrium - Shift in AC

• Starting from a long-run equilibrium, suppose all firms experience a decrease in their average cost (represented by a shift from AC to AC′ in the figure).
• At the current price equilibrium, firms now enjoy positive economic profit, which encourages the entry of additional firms.
• When long-run equilibrium is restored, a typical firm again earns zero profits, but this occurs at the higher long-run equilibrium price.
• The entry of more firms has driven the equilibrium price up!